The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q ?
- (a) p(10⁴ - k²)² = q × 10⁸ ✓ UPSC's answer
- (b) p(10⁴ - k²)² = q × 10⁴
- (c) p(10⁴ - k²) = q × 10⁴
- (d) p(10⁴ - k²) = q × 10⁸
Why the answer is (a)
• An increase by k% multiplies the price by (100 + k)/100.
• A decrease by k% multiplies the price by (100 - k)/100.
• One increase followed by one decrease gives factor ((100 + k)(100 - k))/100² = (10⁴ - k²)/10⁴.
• The sequence has two such increase-decrease cycles, so q = p[(10⁴ - k²)/10⁴]².
• Multiplying both sides by 10⁸ gives p(10⁴ - k²)² = q × 10⁸, which is option (a).
Why the other options are wrong
- (b) p(10⁴ - k²)² = q × 10⁴
- Option (b) uses 10⁴ instead of 10⁸, so it accounts for only one increase-decrease cycle rather than two.
- (c) p(10⁴ - k²) = q × 10⁴
- Option (c) lacks the square on (10⁴ - k²) and uses 10⁴, so it represents only one cycle with the wrong power.
- (d) p(10⁴ - k²) = q × 10⁸
- Option (d) lacks the square on (10⁴ - k²), so it represents only one increase-decrease cycle even though the right side has 10⁸.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.